# Equations Involving Absolute Values

Example 1. Solve |2x-7|=3.

If |a|=3 then either a=3 or a=-3. This means that given equation is equivalent to the set of equations: 2x-7=3,2x-7=-3.

From first equation we find that x=5, from second we find that x=2. Thus, there are 2 solutions: 2 and 5.

Example 2. Solve |2x-8|=3x+1.

We need to consider two cases: 2x-8>=0 and 2x-8<0.

If 2x-8>=0 then |2x-8|=2x-8 and given equation can be rewritten as 2x-8=3x+1. From this we have that x=-9. However, when x=-9 inequality 2x-8>=0 doesn't hold (2*(-9)-8=-26<0), therefore, x=-9 is not root of the equation.

If 2x-8<0 then |2x-8|=-(2x-8) and given equation can be rewritten as -(2x-8)=3x+1. From this we have that x=7/5 . When x=7/5 inequality 2x-8<0 holds (2*(7/5)-8=-26/5<0), therefore, x=7/5 is root of the equation.

Therefore, there is only one root: x=7/5.

Note, that equation of the form |x-a|=b can be also solved geometrically.